z9o8.] THE PHYSICS OF THE EARTH. 189 



P 



K = 



3 r 



. , = :; I p • A.'irr'x" ■ rdx 



9(^o^y • ^ / ^ — sin ^ cos q s.in^ q \ 

 ~ 2(o-,^y \ 2q 3~ /• 



Jf r is expressed in meters, the mean pressure or mean rigidity K 

 comes out in kilograms per square meter. To reduce the result to 

 atmospheres we divide by 10,333. The result for the earth is 

 R = 748,843 atmospheres, about the rigidity of wrought iron. 



This method takes no account of the earth's solid crust, and is 

 therefore too small ; moreover viscosity increases within the earth, 

 owing to the rise of temperature downward. We give hereafter an 

 approximation to the increase of rigidity by determining the mean 

 rigidity of the earth's matter, as distinguished from that of the 

 various layers composing the globe, just found by the above analysis. 



To find the mean rigidity of the earth's matter we must consider 

 not only the pressure but also the density or mass per unit volume 

 of the imprisoned matter in each layer. The result represents a 

 mean rigidity in which every elementary spherical shell composing 

 the globe is allowed a weight proportional to its mass, which is 

 multiplied by the pressure to which it is subjected. 



The theory of the determination of the mean rigidity of the 

 earth's matter is as follows : 



C C" ■> sin {qx) 



P' = /■47r/--,r-;Y/.t'-o- = 47rrV„ p-^-dx—^^-. (7) 



Jo Jo 'i-^ 



Substituting for p its value from (2), we get 



licr^gf ■ r ■ 47rrV^j / T' sin^ {qx)x^dx . ., r'x"' sin {qx)dx 



P 



o;J C'sirf (q-'^Xv-dx . ., r ' x' sm (qx)dx\ ^^; 



3 (S<^)' ■ ^' ■ 4'^^'" • o-o / r'sin^ {qx)qdx . , rx sin (qx)qdx \ 



<T^. / r-'sin^ (qxjqdx . , r\v sm {qx)qdx\ 



- 2(.,^y Hi^^- ^'""d , )-(^) 



^, . , , , . , . . „ sin (qx) — qx cos (qx) 

 The mtegral of this last term is — sin- q ^ , 



The value of the first integral is most conveniently found by quad- 

 rature, table for which is given in A. N., 4104, p. 379. Dividing out 

 the mass, or volume of the sphere by the density, we have 



PROC. AMER. PHIL. SOC. XLVII. 189 M, PRINTED SEPTEMBER 22, I908. 



